1. **Problem Statement:** Sketch and analyze the curves for the following polynomial functions: a) $y = x^2 (x + 2)$ b) $y = x^2 (5 - 2x)$ c) $y = (x + 1)^2 (x - 2)$ d) $y = (x - 2)^2 (10 - 3x)$ 2. **General Approach:** For each polynomial, we will: - Identify the degree and leading term. - Find the roots (x-intercepts) by setting $y=0$. - Determine the multiplicity of each root to understand the behavior at intercepts. - Analyze end behavior based on the leading term. - Sketch the curve accordingly. 3. **Detailed Steps:** ### a) $y = x^2 (x + 2)$ - Expand: $y = x^3 + 2x^2$ - Degree: 3 (cubic), leading term $x^3$ (positive leading coefficient) - Roots: $x=0$ (multiplicity 2), $x=-2$ (multiplicity 1) - Behavior at $x=0$: touches x-axis and turns (because multiplicity 2) - Behavior at $x=-2$: crosses x-axis (multiplicity 1) - End behavior: as $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$ ### b) $y = x^2 (5 - 2x)$ - Expand: $y = 5x^2 - 2x^3$ - Degree: 3, leading term $-2x^3$ (negative leading coefficient) - Roots: $x=0$ (multiplicity 2), $x=\frac{5}{2}$ (multiplicity 1) - Behavior at $x=0$: touches and turns - Behavior at $x=\frac{5}{2}$: crosses - End behavior: as $x \to \infty$, $y \to -\infty$; as $x \to -\infty$, $y \to \infty$ ### c) $y = (x + 1)^2 (x - 2)$ - Expand: $y = (x^2 + 2x + 1)(x - 2) = x^3 - 2x^2 + 2x^2 - 4x + x - 2 = x^3 - x - 2$ - Roots: $x = -1$ (multiplicity 2), $x=2$ (multiplicity 1) - Behavior at $x=-1$: touches and turns - Behavior at $x=2$: crosses - End behavior: as $x \to \infty$, $y \to \infty$; as $x \to -\infty$, $y \to -\infty$ ### d) $y = (x - 2)^2 (10 - 3x)$ - Expand: $y = (x^2 - 4x + 4)(10 - 3x) = 10x^2 - 40x + 40 - 3x^3 + 12x^2 - 12x = -3x^3 + 22x^2 - 52x + 40$ - Roots: $x=2$ (multiplicity 2), $x=\frac{10}{3} \approx 3.33$ (multiplicity 1) - Behavior at $x=2$: touches and turns - Behavior at $x=\frac{10}{3}$: crosses - End behavior: as $x \to \infty$, $y \to -\infty$; as $x \to -\infty$, $y \to \infty$ 4. **Summary:** - Multiplicity 2 roots cause the graph to touch and turn at the x-axis. - Multiplicity 1 roots cause the graph to cross the x-axis. - Leading term determines end behavior. This analysis helps sketch each curve accurately.